Torsion properties of modified diagonal classes on triple products of modular curves

Abstract

Consider three normalised cuspidal eigenforms of weight 2 and prime level p. Under the assumption that the global root number of the associated triple product L-function is +1, we prove that the complex Abel-Jacobi image of the modified diagonal cycle of Gross-Kudla-Schoen on the triple product of the modular curve X 0 (p) is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel-Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow-Heegner points associated with modified diagonal cycles on elliptic curves of prime level with split multiplicative reduction. The approach also works in the case of composite square-free level.